Lucas Pastor November 15 2017 Joint-work with Rmi de Joannis de - - PowerPoint PPT Presentation

Coloring squares of claw-free graphs

Lucas Pastor

November 15 2017 Joint-work with Rémi de Joannis de Verclos and Ross J. Kang

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A (proper) k-coloring of G is an assignement of colors {1, . . . , k} to the vertices of G such that any two adjacent vertices receive a different color.

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A (proper) k-coloring of G is an assignement of colors {1, . . . , k} to the vertices of G such that any two adjacent vertices receive a different color.

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A (proper) k-coloring of G is an assignement of colors {1, . . . , k} to the vertices of G such that any two adjacent vertices receive a different color.

1 1 2 3

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A (proper) k-coloring of G is an assignement of colors {1, . . . , k} to the vertices of G such that any two adjacent vertices receive a different color.

1 1 2 3

The chromatic number, χ(G), is the smallest k such that G is k-colorable.

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A (proper) k-edge-coloring of G is an assignment of colors {1, . . . , k} to the edges of G such that any two adjacent edges (sharing a vertex) receive a different color.

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A (proper) k-edge-coloring of G is an assignment of colors {1, . . . , k} to the edges of G such that any two adjacent edges (sharing a vertex) receive a different color.

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A (proper) k-edge-coloring of G is an assignment of colors {1, . . . , k} to the edges of G such that any two adjacent edges (sharing a vertex) receive a different color.

1 2 2 3

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A (proper) k-edge-coloring of G is an assignment of colors {1, . . . , k} to the edges of G such that any two adjacent edges (sharing a vertex) receive a different color.

1 2 2 3

The chromatic index, χ′(G), is the smallest k such that G is k-edge-colorable.

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Note that in an edge coloring, each color class is a matching.

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Note that in an edge coloring, each color class is a matching.

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Note that in an edge coloring, each color class is a matching.

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Note that in an edge coloring, each color class is a matching.

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Note that in an edge coloring, each color class is a matching.

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But not necessarily an induced matching!

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A strong k-edge-coloring of G is a k-edge-coloring where each color class is an induced matching.

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A strong k-edge-coloring of G is a k-edge-coloring where each color class is an induced matching.

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A strong k-edge-coloring of G is a k-edge-coloring where each color class is an induced matching.

1 2 3 4

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A strong k-edge-coloring of G is a k-edge-coloring where each color class is an induced matching.

1 2 3 4

The strong chromatic index, χ′

s(G), is the smallest k such that

G is strong k-edge-colorable.

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Questions Given a graph G with maximum degree ∆(G).

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Questions Given a graph G with maximum degree ∆(G). χ′

s(G) 6

Questions Given a graph G with maximum degree ∆(G). χ′

s(G) ≤ upper bound 6

Questions Given a graph G with maximum degree ∆(G). lower bound ≤ χ′

s(G) ≤ upper bound 6

Upper bound Pick any edge e, and look at how large can be its neighborhood at distance 2.

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Upper bound Pick any edge e, and look at how large can be its neighborhood at distance 2.

e

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Upper bound Pick any edge e, and look at how large can be its neighborhood at distance 2.

e

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Upper bound Pick any edge e, and look at how large can be its neighborhood at distance 2.

e ∆ ∆

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Upper bound Pick any edge e, and look at how large can be its neighborhood at distance 2.

e ∆ ∆

∆ − 1

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Upper bound Pick any edge e, and look at how large can be its neighborhood at distance 2.

e ∆ ∆

∆ − 1

∆ ∆

∆ − 1

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Upper bound Pick any edge e, and look at how large can be its neighborhood at distance 2.

e ∆ ∆

∆ − 1

∆ ∆

∆ − 1

χ′

s(G) ≤ 2∆(∆ − 1) + 1 = 2∆2 − 2∆ + 1. 7

Lower bound For any even integer ∆ ≥ 2, there exist a graph G of max degree ∆ such that: χ′

s(G) = 5

4∆2.

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Lower bound For any even integer ∆ ≥ 2, there exist a graph G of max degree ∆ such that: χ′

s(G) = 5

4∆2.

8

Lower bound For any even integer ∆ ≥ 2, there exist a graph G of max degree ∆ such that: χ′

s(G) = 5

4∆2.

∆ 2 ∆ 2 ∆ 2 ∆ 2 ∆ 2

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Lower bound For any even integer ∆ ≥ 2, there exist a graph G of max degree ∆ such that: χ′

s(G) = 5

4∆2.

∆ 2 ∆ 2 ∆ 2 ∆ 2 ∆ 2

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Lower bound For any even integer ∆ ≥ 2, there exist a graph G of max degree ∆ such that: χ′

s(G) = 5

4∆2.

∆ 2 ∆ 2 ∆ 2 ∆ 2 ∆ 2

In this graph, any pair of edges is at distance at most 2. There are

5 4∆2 edges in G. 8

Lower bound For any even integer ∆ ≥ 2, there exist a graph G of max degree ∆ such that: χ′

s(G) = 5

4∆2.

∆ 2 ∆ 2 ∆ 2 ∆ 2 ∆ 2

1 4 ∆2

In this graph, any pair of edges is at distance at most 2. There are

5 4∆2 edges in G. 8

Lower bound For any even integer ∆ ≥ 2, there exist a graph G of max degree ∆ such that: χ′

s(G) = 5

4∆2.

∆ 2 ∆ 2 ∆ 2 ∆ 2 ∆ 2

1 4 ∆2 1 4 ∆2 1 4 ∆2 1 4 ∆2 1 4 ∆2

In this graph, any pair of edges is at distance at most 2. There are

5 4∆2 edges in G. 8

Conjecture [Erdős, Nešetřil 1988] The previous example is the worst you can get. In other words: For any graph G, χ′

s(G) ≤ 5 4∆(G)2. 9

Conjecture [Erdős, Nešetřil 1988] The previous example is the worst you can get. In other words: For any graph G, χ′

s(G) ≤ 5 4∆(G)2.

We have an upper bound of 2∆(G)2. Can we do better?

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Conjecture [Erdős, Nešetřil 1988] The previous example is the worst you can get. In other words: For any graph G, χ′

s(G) ≤ 5 4∆(G)2.

We have an upper bound of 2∆(G)2. Can we do better? Theorem [Molloy, Reed 1997] χ′

s(G) ≤ (2 − ǫ)∆(G)2 9

Conjecture [Erdős, Nešetřil 1988] The previous example is the worst you can get. In other words: For any graph G, χ′

s(G) ≤ 5 4∆(G)2.

We have an upper bound of 2∆(G)2. Can we do better? Theorem [Molloy, Reed 1997] χ′

s(G) ≤ (2 − ǫ)∆(G)2

for some constant ǫ = 0.002.

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Conjecture [Erdős, Nešetřil 1988] The previous example is the worst you can get. In other words: For any graph G, χ′

s(G) ≤ 5 4∆(G)2.

We have an upper bound of 2∆(G)2. Can we do better? Theorem [Molloy, Reed 1997] χ′

s(G) ≤ (2 − ǫ)∆(G)2

for some constant ǫ = 0.002. The constant has been improved by Bruhn and Joos in 2015 to ǫ = 0.07.

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Line-graph Given a graph G, the line-graph of G, denoted by L(G), is the graph whose vertices are the edges of G and whose edges are the pairs of adjacent edges of G.

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Line-graph Given a graph G, the line-graph of G, denoted by L(G), is the graph whose vertices are the edges of G and whose edges are the pairs of adjacent edges of G.

e1 e2 e3 e4 e5 e6 G e1 e2 e4 e5 e6 e3 L(G)

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Square graph Given a graph G, the square of G, denoted by G2, is the graph

• btained from G by adding edges between every pair of vertices at

distance 2.

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Square graph Given a graph G, the square of G, denoted by G2, is the graph

• btained from G by adding edges between every pair of vertices at

distance 2.

G

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Square graph Given a graph G, the square of G, denoted by G2, is the graph

• btained from G by adding edges between every pair of vertices at

distance 2.

G G2

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Square graph Given a graph G, the square of G, denoted by G2, is the graph

• btained from G by adding edges between every pair of vertices at

distance 2.

G G2

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Strong coloring

• Coloring the edges of G is equivalent to coloring the vertices
• f L(G).

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Strong coloring

• Coloring the edges of G is equivalent to coloring the vertices
• f L(G).
• The strong coloring of G is equivalent to color G2.

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Strong coloring

• Coloring the edges of G is equivalent to coloring the vertices
• f L(G).
• The strong coloring of G is equivalent to color G2.
• Hence, the strong edge coloring of G is equivalent to color the

vertices of L(G)2.

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Strong coloring

• Coloring the edges of G is equivalent to coloring the vertices
• f L(G).
• The strong coloring of G is equivalent to color G2.
• Hence, the strong edge coloring of G is equivalent to color the

vertices of L(G)2. Molloy and Reed’s theorem Let G be the line-graph of any simple graph, then: χ(G2) ≤ (2 − ǫ)ω(G)2.

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Line-graphs In a line-graph, the neighborhood of any vertex is the union of at most 2 cliques.

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Line-graphs In a line-graph, the neighborhood of any vertex is the union of at most 2 cliques.

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Line-graphs In a line-graph, the neighborhood of any vertex is the union of at most 2 cliques.

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Line-graphs In a line-graph, the neighborhood of any vertex is the union of at most 2 cliques.

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Line-graphs In a line-graph, the neighborhood of any vertex is the union of at most 2 cliques. The class of graphs having this property is the class of quasi-line graphs.

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Quasi-line graphs In a quasi-line graph, the neighborhood of any vertex cannot have 3 pairwise non-adjacent vertices.

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Quasi-line graphs In a quasi-line graph, the neighborhood of any vertex cannot have 3 pairwise non-adjacent vertices.

claw

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Quasi-line graphs In a quasi-line graph, the neighborhood of any vertex cannot have 3 pairwise non-adjacent vertices.

claw

The class of graphs having this property is the class of claw-free graphs.

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line-graph

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line-graph quasi-line

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line-graph quasi-line claw-free

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line-graph quasi-line claw-free Molloy and Reed Molloy and Reed

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line-graph quasi-line claw-free Molloy and Reed Molloy and Reed Us

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Theorem [de Joannis de Verclos, Kang, P.] There is an absolute constant ǫ > 0 such that, for any claw-free graph G: χ(G2) ≤ (2 − ǫ)ω(G)2

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Theorem [de Joannis de Verclos, Kang, P.] There is an absolute constant ǫ > 0 such that, for any claw-free graph G: χ(G2) ≤ (2 − ǫ)ω(G)2 Roadmap

• 1. From claw-free to quasi-line graphs.

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Theorem [de Joannis de Verclos, Kang, P.] There is an absolute constant ǫ > 0 such that, for any claw-free graph G: χ(G2) ≤ (2 − ǫ)ω(G)2 Roadmap

• 1. From claw-free to quasi-line graphs.
• 2. From quasi-line graphs to line-graphs of multigraphs.

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Theorem [de Joannis de Verclos, Kang, P.] There is an absolute constant ǫ > 0 such that, for any claw-free graph G: χ(G2) ≤ (2 − ǫ)ω(G)2 Roadmap

• 1. From claw-free to quasi-line graphs.
• 2. From quasi-line graphs to line-graphs of multigraphs.
• 3. Prove the theorem for line-graphs of multigraphs.

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Second neighborhood The second neighborhood of v, denoted by N2

G(v), is the set of

vertices at distance exactly two from v, i.e. N2

G(v) = NG2(v) \ NG(v). 17

Second neighborhood The second neighborhood of v, denoted by N2

G(v), is the set of

vertices at distance exactly two from v, i.e. N2

G(v) = NG2(v) \ NG(v).

The square degree of v, denoted by degG2(v), is equal to degG(v) + |N2

G(v)|. 17

Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V (G) with degG2(v) ≤ ω(G)2 + (ω(G) + 1)/2 whose neighborhood is a clique of (G \ v)2.

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Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V (G) with degG2(v) ≤ ω(G)2 + (ω(G) + 1)/2 whose neighborhood is a clique of (G \ v)2.

• 1. The proof is by induction on |V (G)|.

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Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V (G) with degG2(v) ≤ ω(G)2 + (ω(G) + 1)/2 whose neighborhood is a clique of (G \ v)2.

• 1. The proof is by induction on |V (G)|.
• 2. Note that (G \ v)2 = G2 \ v.

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v

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v

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v

19

19

19

v

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v

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v clique in G2 − v

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maybe not in (G \ v)2

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Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V (G) with degG2(v) ≤ ω(G)2 + (ω(G) + 1)/2 whose neighborhood is a clique of (G \ v)2.

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Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V (G) with degG2(v) ≤ ω(G)2 + (ω(G) + 1)/2 whose neighborhood is a clique of (G \ v)2. Claim If NG(v) is not a clique of (G \ v)2 then NG(v) is the union of two cliques.

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Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V (G) with degG2(v) ≤ ω(G)2 + (ω(G) + 1)/2 whose neighborhood is a clique of (G \ v)2. Claim If NG(v) is not a clique of (G \ v)2 then NG(v) is the union of two cliques.

v

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Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V (G) with degG2(v) ≤ ω(G)2 + (ω(G) + 1)/2 whose neighborhood is a clique of (G \ v)2. Claim If NG(v) is not a clique of (G \ v)2 then NG(v) is the union of two cliques.

v x y

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Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V (G) with degG2(v) ≤ ω(G)2 + (ω(G) + 1)/2 whose neighborhood is a clique of (G \ v)2. Claim If NG(v) is not a clique of (G \ v)2 then NG(v) is the union of two cliques.

v x y d(x, y) ≥ 3

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Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V (G) with degG2(v) ≤ ω(G)2 + (ω(G) + 1)/2 whose neighborhood is a clique of (G \ v)2. Claim If NG(v) is not a clique of (G \ v)2 then NG(v) is the union of two cliques.

v x y d(x, y) ≥ 3

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Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V (G) with degG2(v) ≤ ω(G)2 + (ω(G) + 1)/2 whose neighborhood is a clique of (G \ v)2. Claim If NG(v) is not a clique of (G \ v)2 then NG(v) is the union of two cliques.

v x y d(x, y) ≥ 3

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Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V (G) with degG2(v) ≤ ω(G)2 + (ω(G) + 1)/2 whose neighborhood is a clique of (G \ v)2. Claim If NG(v) is not a clique of (G \ v)2 then NG(v) is the union of two cliques.

v x y d(x, y) = 3

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Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V (G) with degG2(v) ≤ ω(G)2 + (ω(G) + 1)/2 whose neighborhood is a clique of (G \ v)2. Claim If NG(v) is not a clique of (G \ v)2 then NG(v) is the union of two cliques.

v x y d(x, y) = 3 z

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Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V (G) with degG2(v) ≤ ω(G)2 + (ω(G) + 1)/2 whose neighborhood is a clique of (G \ v)2. Claim If NG(v) is not a clique of (G \ v)2 then NG(v) is the union of two cliques.

v x y d(x, y) = 3 z

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Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V (G) with degG2(v) ≤ ω(G)2 + (ω(G) + 1)/2 whose neighborhood is a clique of (G \ v)2. Claim If NG(v) is not a clique of (G \ v)2 then NG(v) is the union of two cliques.

v x y d(x, y) = 3 z

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Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V (G) with degG2(v) ≤ ω(G)2 + (ω(G) + 1)/2 whose neighborhood is a clique of (G \ v)2. Claim If NG(v) is not a clique of (G \ v)2 then NG(v) is the union of two cliques.

v x y d(x, y) = 3 z

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Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V (G) with degG2(v) ≤ ω(G)2 + (ω(G) + 1)/2 whose neighborhood is a clique of (G \ v)2. Claim If NG(v) is not a clique of (G \ v)2 then NG(v) is the union of two cliques.

v x y d(x, y) = 3 z

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Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V (G) with degG2(v) ≤ ω(G)2 + (ω(G) + 1)/2 whose neighborhood is a clique of (G \ v)2. Claim If NG(v) is not a clique of (G \ v)2 then NG(v) is the union of two cliques.

v x y d(x, y) = 3 z

20

Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V (G) with degG2(v) ≤ ω(G)2 + (ω(G) + 1)/2 whose neighborhood is a clique of (G \ v)2. Claim If NG(v) is not a clique of (G \ v)2 then NG(v) is the union of two cliques.

v x y d(x, y) = 3 z

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Conclusion

• Our constant can be improved by using Bruhn and Joos

method.

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Conclusion

• Our constant can be improved by using Bruhn and Joos

method.

• The conjecture for bipartite graphs is χ′

s(G) ≤ ∆(A)∆(B). 21

Thank you for your attention.

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